We summarize the works on the parabolic Dirichlet boundary control problems
briefly as follows. Using an integral representation formula derived by the
semigroup theory (cf. [1]} and
[2]}), Lasiecka [3]}, [4]} analyzed
spatial Galerkin approximations of an optimal Dirichlet boundary control problem
and a time optimal Dirichlet boundary control problem for the parabolic
equations. Kunisch and Vexler [5]} analyzed constrained Dirichlet
boundary control problems for a class of parabolic equations and derived the
convergence of the PDAS strategy for two Dirichlet boundary control problems.
Applying the Robin penalization method to a Dirichlet boundary control problem
for a parabolic equation with a final observation, Belgacem et
al. [6]} obtained a penalized Robin boundary control problem.
Gong et al. [7]} analyzed a finite element approximation of a
Dirichlet boundary control for a parabolic equation, where the variational
discretization approach [8]} was used and the state equation was
discretized by the usual \( H^1 \) -conforming \( P1 \) -element in space and
discretized by the \(\text{dG}(0)\)  scheme in time. Recently, Gong and Li
[9]} improved the spatial accuracy derived in [7]},
using the maximal \( L^p \) -regularity theory. We note that, for the state
equation with rough Dirichlet boundary data,
[3]}, [4]} used the semigroup theory to define the
solution whereas [7]}, [9]}, [5]} used the transposition
technique to define the solution (called the very weak solution).
